TY - JOUR

T1 - On Composite Quantum Hypothesis Testing

AU - Berta, Mario

AU - Brandão, Fernando G.S.L.

AU - Hirche, Christoph

N1 - Funding Information: We thank an anonymous referee for extensive feedback and pointing out detailed solutions to multiple errors in previous versions of this manuscript. This work was completed prior to MB and FB joining the AWS Center for Quantum Computing. CH acknowledges support from the VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059), the Spanish MINECO, project FIS2013-40627-P, FIS2016-80681-P (AEI/FEDER, UE) and FPI Grant No. BES-2014-068888, as well as by the Generalitat de Catalunya, CIRIT project no. 2014-SGR-966.

PY - 2021/6/10

Y1 - 2021/6/10

N2 - We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ρ⊗n against convex combinations of quantum states σ⊗n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.

AB - We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ρ⊗n against convex combinations of quantum states σ⊗n can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.

UR - http://www.scopus.com/inward/record.url?scp=85107461338&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-04133-8

DO - 10.1007/s00220-021-04133-8

M3 - Article

AN - SCOPUS:85107461338

VL - 385

SP - 55

EP - 77

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -